To further study the SC properties of La\(_2\) Rh\(_{3+\delta }\) Sb\(_4\) , we turn to specific heat measurements, the results of which are summarized in Fig. REF . At 200 K, the total specific heat \(C\)  is about 206 J/mol\(\cdot \) K, close to the classic limit 3\(NR\)  = 224.5 J/mol\(\cdot \) K given by Dulong-Petit's law[1]}, where \(N\)  = 9 is the number of atoms in a formula unit of stoichiometic La\(_2\) Rh\(_3\) Sb\(_4\) , and \(R\)  = 8.314 J/mol\(\cdot \) K is the ideal gas constant. The low temperature specific heat conforms to \(C(T)=\gamma _n T+\beta T^3\) , where the first and second terms are contributions from electrons (\(C_{el}\) ) and phonons (\(C_{ph}\) ). This is better expressed by the \(C/T\)  vs. \(T^2\)  plot inset to Fig. REF . This fit leads to the normal-state Sommerfeld coefficient \(\gamma _n=19.45\)  mJ/mol\(\cdot \) K\(^2\) , and \(\beta =\)  1.21 mJ/mol\(\cdot \) K\(^4\) , and enables us to estimate the Debye temperature, \(\Theta _D=(\frac{12\pi ^4NR}{5\beta })^{1/3}=244\)  K. Note that the rapid increase of \(C/T\)  for \(T\rightarrow 0\)  shown in the upper inset of Fig. REF  arises from the nuclear Schottky anomaly (\(C_{Sch}\) ) and can be properly subtracted by fitting \(C_{Sch}\)  to a \(1/T^2\)  law[2]}. We then obtain \(C_{el}\)  by subtracting both \(C_{ph}\)  and \(C_{Sch}\)  from the total \(C(T)\) , as shown in the lower inset to Fig. REF . The anomaly in specific heat due to the SC transition now can be clearly seen near 0.65 K, and the jump of specific heat at \(T_c\)  is estimated \(\Delta C/T|_{T_c}=6.5\)  mJ/mol\(\cdot \) K\(^2\) . Below \(T_c\) , the electronic specific heat consists of two parts, viz. \(C_{el}(T)=\gamma _0 T+C_{sc}(T)\) , where \(\gamma _0\)  is the residual Sommerfeld coefficient, and \(C_{sc}\)  describes the superconducting part of specific heat. We find that \(C_{sc}\)  agrees well to the \(s\) -wave model[3]}, viz. \(C_{sc}/T\propto T^{-5/2}\exp {(-\Delta _0/T)}\) , in which \(\Delta _0\)  characterizes the superconducting gap. This fit gives rise to \(\gamma _0=15.0\)  mJ/mol\(\cdot \) K\(^2\)  and \(\Delta _0=\)  1.1 K.
This analysis further leads to two important ratios, \(\frac{\Delta C}{(\gamma _n-\gamma _0)T_c}=1.46\)  and \(\Delta _0/T_c=\)  1.69. These values respectively are close to 1.43 and 1.76 as predicted by the BCS's theory[4]}, [5]}. La\(_2\) Rh\(_{3+\delta }\) Sb\(_4\) , therefore, is likely a conventional \(s\) -wave superconductor.
